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Flow velocity measurements using shock waves - the swing probe Strachan, James Douglas 1972

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11 ^ £ 2 FLOW VELOCITY MEASUREMENTS USING SHOCK WAVES « THE SWING PHOBE by James D. Strachan B.Sc, University of B r i t i s h Columbia, 1968 M . S c , U n i v e r s i t y of B r i t i s h Co lumb ia , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada •6 - 2 -ABSTRACTt The theory of shock propagation into inhomo-geneous media has been extended to include continuous va r i a t i o n s i n the i n i t i a l pressure, p a r t i c l e v e l o c i t y , and energy sources appearing at the f r o n t . The one dimensional equations have been developed to allow a shock to be used as probe„ Shock waves which c o l l i d e with unknown gas or plasma flow f i e l d s s u f f e r a change i n v e l o c i t y . Pressure, density, p a r t i c l e v e l o c i t y , and l o c a l energy input at the edge of an unknown flow can be determined from the measurement of the relevant fronts upon c o l l i s i o n of the probing shock with the edge of the unknown flow. The steady v a r i a t i o n of the v e l o c i t y of strong probing shocks reveals d e t a i l s of the l o c a l flow v e l o c i t y and density d i s t r i b u t i o n s inside the unknown flow f i e l d . The flow v e l o c i t y of a T-tube plasma has been measured with t y p i c a l accuracies of .1 Km/sec which Is about 10% of the v e l o c i t y of the probing shock. The i n t e r p r e t a t i o n of the r e s u l t s i n v i t e s a comparison of the T-tube flow with a breakdown wave In the high v e l o c i t y regime and with a r a d i a t i o n driven shock i n the low v e l o c i t y regime f - 3 -TABLE OF CONTENTS page section 2. Abstract 3. Table of contents k. L i s t of variables 6. Aknowledgements ?. Introduction 11. Chapter I - Shocks with external sources of mass, momentum, and energy 15. I 1) Solution with source terms 21 . 1 2 ) E f f e c t i v e adiabatic constant 2^. Chapter II - Shock propagation into inhomogeneous media 25. II 1) Discontinuous inhomogeneous regions 3 2 . II 2) Continuous inhomogeneous regions &7. II 3) Experimental a p p l i c a t i o n of the shock front equation 55. Chapter III - Theory of the SWING probe 57. I l l 1) Probing discontinuous unknown flows 5 8 . I l l 2) Probing continuous unknown flows 61 . I l l 3) Properties of the SWING probe 6 6 . Chapter IV - P a r t i c l e V e l o c i t y measurements of a T-tube flow using the SWING probe 6 9 . IV 1) Description of the experiment 7 2 . IV 2) The high v e l o c i t y regime 7 8 . IV 3) Models of the T-tube flow 8 0 . IV h) The low v e l o c i t y regime 9 1 . Summary 9 3 . References 9 6 . Appendix A - Apparatus /*^ v\' 104. Appendix B - Another de r i v a t i o n of v ^ u ] LIST OF VARIABLESi CHAPTER I •f - d i s t r i b u t i o n function X - time - component of v e l o c i t y * i - 1 component of p o s i t i o n C - 1 t h component of external force term V - shock v e l o c i t y i n the lab frame-AT - p a r t i c l e v e l o c i t y i n the shock frame S - density - pressure 6' - dimensionless mass source term c - speed of sound tl - Mach number f ' - dimensionless momentum source term i - current (k - magnetic f i e l d 0 - e f f e c t i v e adiabatic constant W- dimensionless energy source term - p a r t i c l e v e l o c i t y i n the lab frame -A- - enthalpy ^ - i n t e r n a l energy cpS#- s p e c i f i c heats r*. - number of degrees of freedom CHAPTER II \ - shock d r i v i n g force W*- e f f e c t i v e energy source term W - energy source term ^ - momentum source term ft - external source parameter t - shock pressure r a t i o tSf.) U~V > W ' ' d i m e n s i o n l e s s c o e f f i c i e n t s - electron number density Bji - i o n i z a t i o n energy T - temperature ^ - i n t e n s i t y of absorbed external energy 3^- i n t e n s i t y of absorbed external energy required to sustain a Chapman-Jouguet detonation at a s p e c i f i c v e l o c i t y CHAPTER I I I <s* - e f f e c t i v e dimensionless c o e f f i c i e n t which i s an approximate dV - 5 -LIST OF VARIABLES t CHAPTER IV Jtr - Alfven v e l o c i t y Q) - inverse den'sity r a t i o € - dimensionless parameter describing transverse i o n i z i n g shocks. -5 , -, a) LIST OF FIGURES page nu: 13 1 16 2 17 3 18 4' 22 5 29 6 35 7 36 8 37 9 38 10 39 11 42 12 44 13 45 14 48 16 52 17 53 18 62 19 68 20 73 21 74 22 75 23 77 24 82 25 83 26 84 27 85 28 88 29 97 30 102 31 103 32 105 15 mber caption l i c i e n t OgL)' : i c i e n t (Js \' : i c i e n t IsuJ Shock wave in shock frame Density r a t i o as a function of external sources Pressure r a t i o as a function of external sources P a r t i c l e v e l o c i t y as a function of external sources g(-P\T*) for argon plasma i n thermal equilibrium general i n t e r a c t i o n scheme^ Dimensionless c o e f f i Dimensionless coeff: Dimensionless c o e f f i c i e n t v"(.su^ Dimensionless c o e f f i c i e n t (|^ ) C o e f f i c i e n t s i n strong shock approximation Interaction scheme for the absorption of radiation at a shock /*ay\» Dimensionless coeff i c i e n t Cfx) /^\/s\' Dimensionless c o e f f i c i e n t v ) Shock propagation through a known rar e f a c t i o n wave Shock v e l o c i t y through the known rarefac t i o n wave Shock v e l o c i t y through several r a r e f a c t i o n waves Approximate shock c o e f f i c i e n t tf* Experimental setup Front v e l o c i t i e s for the c o l l i s i o n s i n the low pressure regime Flow v e l o c i t y immediately behind the T-tube front External source parameter for the T-tube flow External source parameter for the secondary waves A (distance,time) diagram of the T-tube flow at high pressures Smear pictures of the c o l l i s i o n s at high pressures Front v e l o c i t i e s for the high pressure c o l l i s i o n s Flow v e l o c i t i e s at high pressures P a r t i c l e v e l o c i t y i n the continuous region General layout of the smear camera Construction d e t a i l s of the discharge chamber Capacitor bank for the T-tube . / Interaction scheme for the second derivation of L ^ u J - 6 -ACKNOWLEDGEMENTS» I wish to thank my supervisor, Dr. B. Ahlborn f o r his enthusiastic, guidance through the course of t h i s work. My committee members distinguished themselves with u s e f u l l comments and i n t e r e s t i n g ideas i n our meetings, so I thank Dr. A. J . Barnard, Dr. F.L. Curzon, and Dr. I. Gartshore. As well, I have a s p e c i a l thankyou f o r Dr. R. Nodwell who was my supervisor while Dr. Ahlborn was on s a b a t i c a l . I t has been a pleasure associating with the f r i e n d l y and stimulating people comprising the plasma physics group. In p a r t i c u l a r , I have had many h e l p f u l discussions with Dr. J.P. Huni, Mr. P. Redfern, and Dr. R.C.Cross. F i n a n c i a l support from the National Research Council i s also g r a t e f u l l y acknoweldged. INTRODUCTIONi When the v e l o c i t y of a f l u i d "becomes comparable with or exceeds that of sound, effects due to the compres s i b i l i t y of the f l u i d become important. One of the most d i s t i n c t i v e features of supersonic flow i s that shock waves (strong compression waves) can occur i n i t , "In seeking to produce and understand high temperature pl a s -mas, i t i s necessary to probe deeply into the physics of shock waves." Any time that "a large concentration of energy i n a continuous medium i s suddenly released, I t w i l l spread into the surrounding medium and at- i t s forefront w i l l be a shock wave. The shock wave heats, compresses, and accelerates the medium through which i t propagates. Strong shock waves w i l l ionize the gas and the r e s u l t i n g plasma can i n t e r a c t with em f i e l d s , or the shock may propagate through an already e x i s t i n g plasma^ heating and compressing i t further. A shock wave is an abrupt but continuous change i n the state 1 of the medium," Shocks occur i n many laboratory devices such as simple shock tubes where the released energy i s the p o t e n t i a l (mechanical) energy of a high pressure d r i v i n g gas; f a s t magnetic compression devices (eg theta pinches, z pinches, andplasma focus) where the d r i v i n g energy i s electromagneticj and plasma sources (eg l a s e r sparks, and plasma guns). - 8 -"Shock waves occur i n many natural circumstances. They emanate from tiny sparks and bolts of l i g h t n i n g . Shock waves are produced by s a t e l l i t e s and meteorites as they plunge into the earth's atmosphere, and a shock preceeds the earth as i t moves through the solar wind. Shock waves propagate through the hydrogen of i n t e r s t e l l a r space, driven by the r a d i a t i o n from newborn s t a r s . Nova and supernova, among the most v i o l e n t events i n nature, consist of strong shock waves propagating outward through the plasma of a s t a r a f t e r the sudden release of a large amount of energy at i t s center. Hydrogen clouds i n i n t e r g a l a c t i c space sometimes c o l l i d e and sometimes are captured by the g r a v i t a t i o n a l f i e l d of galaxies, and during the i n t e r a c t i o n very strong shocks are produced." 1 As evident from the above paragraphs by Chu and Gross 1, shock waves encompass a great many phenomena both inside and outside the laboratory. Not s u r p r i s i n g l y , a complete understanding of the phenomena relevent to shock waves must involve a l l of physics . Of course, our s p e c i f i c topics i n t h i s thesis are somewhat narrower and deal e s s e n t i a l l y with the macroscopic i n t e r a c t i o n of the shock with the medium through which i t propagates. The propagation of shocks into uniform gases i s well 1-8 understood , but e f f e c t s due to the l o c a l v a r i a t i o n s i n the medium and external sources has not yet been -9-completely described. A repercussion of the study of t h i s problem has been the developement of a shock probing technique which uses known shocks to measure flow v e l -o c i t i e s i n unknown flows. However i n motivating the work fo r the reader i t i s clea r e r to present a l l the material as part of the answer to the questioni "How can shock waves be used to probe a flow f i e l d i n order to measure flow v e l o c i t i e s ? " In Chapter I I , c a l c u l a t i o n s are presented which describe the influence of l o c a l v a r i a t i o n s i n the medium and external force f i e l d s and energy sources on an established shock wave. These ca l c u l a t i o n s are a generalization of work by C h i s n e l l ^ . Two sets of equations are developed to describe shock propagation across both discontinuous and continuous changes i n the f l u i d para-meters. I t turns out that the shock v e l o c i t y i s c l o s e l y r e l a t e d to the l o c a l p a r t i c l e v e l o c i t y of the medium so that i n Chapter III the two sets of equations are re-examined from the point of view of i n f e r r i n g the flow v e l o c i t y from observations of changes i n a shock v e l o c i t y through an unknown region. This set of equations forms the basis of the measurement* of flow v e l o c i t i e s using shock waves (SWING probe). An a p p l i c a t i o n of t h i s probing technique to study a T-tube flow f i e l d i s presented i n Chapter IV (the apparatus i s described i n the appendix). The SWING probe y i e l d s p a r t i c l e v e l o c i t i e s - 1 0 -to the order of the accuracy which the shock front v e l o c i t y can be measured. The SWING probe represents 10 an Improvement over e x i s t i n g methods i n accuracy and i n general a p p l i c a b i l i t y (since the SWING probe should be useful over a wide range of flow v e l o c i t i e s ) . The standard knowledge on which t h i s work i s based i s sum-marized i n Chapter I. The jump equations are put i n a form which i s convenient f o r the calculations i n the remaining chapters. CHAPTER I SHOCKS WITH EXTERNAL SOURCES OF MASS, MOMENTUM, AND ENERGY In t h i s chapter, solutions to the general jump equations are discussed. These solutions are not new but the approach i s unusual since we consider the external sources to be undefined parameters i n a flow with a given front v e l o c i t y . Of course, i n r e a l i t y the external sources govern the complete behaviour of the flow and thus uniquely define the front v e l o c i t y , however an-t i c i p a t i n g the method of a p p l i c a t i o n of the SWING probe, we presume that the front v e l o c i t y i s the only measure-able quantity of the flow. In gas dynamics, flows are usually described by the Navier-Stokes equations. Any shocks present i n the flow must, therefore, also be contained i n the solutions of the Navier-Stokes equations, under approp-r i a t e i n i t i a l and boundary conditions. However, solutions to the Navier-Stokes equations are d i f f i c u l t to obtain, and such a complete d e s c r i p t i o n cannot usually be expected* But i f the t y p i c a l scale of a problem i s large i n comparison with the t r a n s i t time of a f l u i d p a r t i c l e through the shock thickness, then we may analyze the flow i n two stages $ (1) drop the viscous and thermal conduction terms In the Navier-Stokes equations to obtain the simpler Euler equations, and solve these under correspondingly simpler i n i t i a l and boundary conditionsi the shock waves -12-then appear as d i s c o n t i n u i t i e s . (2) Derive appropriate equations v a l i d inside the t r a n s i t i o n regions from the Navier-Stokes equations by changing the length scales appropriately; these are the shock structure equations. I f the thickness of the shock wave becomes so small as to be comparable to the mean free path, the Navier-Stokes equations become i n v a l i d i n describing the flow In the shock 6 One must then use the Boltzmann equation (which i s highly i n t r a c t a b l e ) or s i m i l a r k i n e t i c equations f o r the d e s c r i p t i o n of the shock structure. There are three l e v e l s of descriptions (1 )nondlssipatlv©r(,discontinuity) des c r i p t i o n , (2) d l s s i p a t l v e Navier-Stokes or shock structure equations, and (3) k i n e t i c theory. Our approach i n t h i s thesis i s e s s e n t i a l l y the nondissipative one. The " d i s c o n t i n u i t y " conservation equations are e a s i l y obtained d i r e c t l y from the Boltzmann equation (1) by multiplying i t with the three 2 factors l,v,and v and integ r a t i n g over v e l o c i t y space. The c o l l i s i o n i n t e g r a l s disappear since the above multiples are conserved i n a c o l l i s i o n . The r e s u l t i s three d i f f e r e n t i a l equations featuring three external where -f i s the d i s t r i b u t i o n function -13-d i r e c t i o n o f p r o p a g a t i o n . v i n l a b frame v 2 «9- * — v l shock heated downstream gas gas f r o n t v i s the p a r t i c l e v e l o c i t y i n the shock frame V i s the f r o n t v e l o c i t y i n the l a b frame u i s the p a r t i c l e v e l o c i t y i n the l a b frame u=v-V F i g . 1 Shock Wave i n Shock Frame source terms. These source terms are i n t e r r e l a t e d since they are derived from the external source R i # We are considering a shock to be a discontinuous Jump i n pressure -p* , density ? , and p a r t i c l e v e l o c i t y U , which, i n the frame of the shock fr o n t , separates uniform equilibrium conditions of subsonic flow from uniform equilibrium conditions of supersonic flow ( F i g . 1 ) . The basic equations used to r e l a t e the f l u i d parameters across a shock are the steady-state conservation equations (2)-(^.)» derived from the three d i f f e r e n t i a l equations. The basic conservation of mass i s i (2) where Q i s an external source term corresponding to the mass added to or removed from the f r o n t . U i s treated as a known parameter of the flow and i s dimensionless f o r convenience i n wri t i n g the f i n a l s o l u t i o n . ( Q'* AO -14-means that the front receives as much mass from the external source as i t sweeps up). Mass source terms can occur i n thermonuclear (fusion) detonations i n which mass i s destroyed i n the front or i n very turbulent shocks where mass i s transported r i g h t to the f r o n t from behind the shock heated gas. The basic conservation of momentum i s §,/v-* + y&, 6 + | , n V ) = (3) where tl i s the Mach number = ^ / c , j C, i s the speed of sound; and F i s a dimensionless external force term which supplies momentum to the f r o n t . Such a term exists i n MHD and i o n i z i n g shocks as the _£XB force due to external magnetic f i e l d s . The basic conservation of energy i s where ^ i s the e f f e c t i v e adiabatic constant and W i s a dimensionless energy source term (normalized by the k i n e t i c energy swept up by the f r o n t ) . External energy sources occur i n r a d i a t i o n fronts,, MHD and i o n i z i n g shocksj l a s e r sparks» and quite generally are the causes of many discontinuous fronts i n physics. Since fronts with energy sources are common, the e f f e c t of energy sources has been studied more and i s consequently better understood than the e f f e c t s of force and mass sources. Thus the d e s c r i p t i o n of shocks having a l l three external sources -15-l s most conveniently described by analogy with fronts which have onlji energy sources. 1 1 ) SOLUTION WITH SOURCE TERMS The exact s o l u t i o n of equations (2)-(*0 are possible using simple algebraj ) Equations ( 5 ) - ( 7 ) are of quite general a p p l i c a -b i l i t y and can be used to describe many phenomena inv o l v i n g the compressive nature of f l u i d s . S p e c i f i c instances are reduced to considerations of the r e l a t i o n s h i p s between the various external source terms. For instance i f F'~ 0'- Vj'- Q t then (2)-(*0 reduce to the usual Rankine-Hugoniot equations f o r adiabatic shocksi i f f'sO'^ Q and i n the strong shock l i m i t ( M V> 1 ) , W - r T 7 * then the wave i s a Chapman-Jouguet det-onation. S i m i l a r l y , a complete d e s c r i p t i o n of a l l external source terms ( i f possible) plus a measurement of the front v e l o c i t y w i l l y i e l d the jump conditions across any f r o n t . I t i s also of some value to examine q u a l i t a -t i v e l y the e f f e c t of each of the external source terms on the jump equations. Figures (2)-(b) indicate the values f o r -16-F i g . 2 D e n s i t y R a t i o as a F u n c t i o n o f E x t e r n a l Sources Chapman /• Jouguet d e t o n a t i o n -0.5 d i m e n s i o n l e s s e x t e r n a l source term -17-F i g . 3 P r e s s u r e R a t i o as a F u n c t i o n of E x t e r n a l Sources d i m e n s i o n l e s s e x t e r n a l s o u r c e term -18-F i g . 4 P a r t i c l e V e l o c i t y as a F u n c t i o n o f E x t e r n a l Sources M= 10.0 g =g„= 1.67 d i m e n s i o n l e s s e x t e r n a l s o u r c e term -19-^~ , » and W, f o r a front i n Argon having a v e l o c i t y of 3 . 2 Km/sec (M=10.0) f o r which the simp l i f y i n g assumption J X»"Y i s made. The curves p'-O'^O • W as the parameter corresponds to the common case of fronts with only an energy source term and the curve can be c l a s s i f i e d according to various experimental phenomena which have been observed. When W ^ O , then energy i s l o s t from the front and the wave can be compared to a piston driven shock which i s very luminous and loses r a d i a t i v e energy. When V/=o » t n ® shock i s adiabatic and i s driven by a simple p i s t o n . When 0 < \^/^X^T~i , the piston driven shock i s supplied external energy which helps drive the shock and the wave i s l i k e an overcompressed detonation When W » , the wave i s a Chapman-Jouguet detonation and i s supported s o l e l y by the energy released at the fr o n t . The exhaust v e l o c i t y A£ i s equal to the speed of sound of the heated gas (Jouguet condition) so that the wave supports i t s e l f without any piston d r i v i n g . The remaining portion of the curve corresponds to a breakdown wave - y , X J-and occurs when the source of energy i s driven f a s t e r than the gas can expand ( Cx<aJ^ ) . The flow i s then c h a r a c t e r i s t i c of the external behaviour of the energy source rather than the act u a l f l u i d behaviour An example of a breakdown wave would be the wave formed by a serie s of independent spark gaps arranged to f i r e i equentlally down a tube. When the f i r s t gap i s discharged -20-a b l a s t wave i s formed which propagates towards the next spark gap. If the next spark gap f i r e s before the b l a s t wave from the f i r s t reaches i t , then the properties o f the heated gas depend only on the l o c a l discharge properties. I f the distance between the sparks i s i n f i n i t e s s i m a l l y small then one macroscopic wave propagates (whose v e l o c i t y depends on the t r i g g e r i n g of the spark gaps) as a break-down wave. The other curves with r and U as parameters do not correspond to known phenomena but do i l l u s t r a t e the e f f e c t of the external source terms. Comparing these fronts to an adiabatic shock t r a v e l l i n g at the same speed y i e l d s i D V o F'>0 QfliftftAflC Swot* As well, one expects that the jump equations f o r any s p e c i f i c phenomena w i l l be described by a s i m i l a r curve with the relationships between the external sources defining i t s exact shape. There ought to be a generalized Jouguet condition f o r any phenomena corresponding to the square root i n equation (5) being zero or -21-a J « O (8) I t i s possible that such a condition may define the naturally-occurring (self-supporting) jump conditions as i t does f o r the well known detonation case and f o r normal i o n i z i n g s h o c k s 1 2 . I .2) EFFECTIVE ADIABATIC CONSTANT parameters without any approximations provided thermal equilibrium may be assumed1-^. The above d e f i n i t i o n f o r the e f f e c t i v e adiabatic constant i s equivalent to defining as a known parameter since i t can be calculated i t e r a t i v e l y as a function o£ the equilibrium pressure and temperature using the i o n i z a t i o n and e x c i t a t i o n properties of the gas. For example, the Q value f o r Argon is p l o t t e d i n 14 figure 5 « The advantage of using the e f f e c t i v e adiabatic constant i s that i t separates the thermodynamic c a l c u l a -tions from the hydrodynamic c a l c u l a t i o n s . Therefore equations (2)-(4) are s u f f i c i e n t (using a curve l i k e f i g u r e 5 r e c u r s i v e l y ) to describe exactly the equilibrium properties of discontinuous fronts i n f l u i d dynamics. the enthalpy as i s treated - 2 2 -Notice i n figure^ 5 that g d i f f e r s from If when the temperature becomes large enough"so that i o n i z a t i o n and d i s s o c i a t i o n (non i d e a l gas behaviour) occur. The adiabatic constant Y i s an important thermostatic property of gases. In c l a s s i c a l thermodynamics is defined as the r a t i o - i — | — i — i i 1 1 1 1 1 1 1 i | ARGON g(p,T) for argon plasma i n t l iermal equi l i s p e c i f i c heats. The value of t h i s F i g . 5 d e f i n i t i o n arises from the many related equations which can be written once one assumes an i d e a l gas (Boyles Law) such as V =- ~ ; Y = ; ^ -where X i s the enthalpy, U i s the Internal energy, and /w i s the number of degrees of freedom, using the eq u i p a r t i t i o n of energy theorem. I t turns out that f o r many processes involving gases at r e l a t i v e l y low pressures and temperatures, the i d e a l gas law may be used to a high degree of accuracy. The phys i c a l Importance of Y i s two-f o l d i i t s value f o r a gas provides evidence concerning the number of degrees of freedom of the molecules c o n s t i t u t i n g the gas; and the adiabatic equation of a gas i s con-veniently expressed i n terms of Y , ( ^ ^ C ' ^ \ _ y or f o r an i d e a l gas -4ft ? = constant, i f ^ i s constant, i n an adi a b a t i c - q u a s l s t a t i c process. When one t r i e s to appljt these concepts to high temperature - 2 3 -gases and plasmas (where now the id e a l gas assumption does not apply) the subdefinitions no longer a l l agree. However f o r our work here i t i s s u f f i c i e n t to only JLi discuss g « /{J ( the e f f e c t i v e "enthalpy" constant), since t h i s i s the only d e f i n i t i o n of Y which appears i n the conservation equations. Therefore, f o r applications to gas dynamics, the c o e f f i c i e n t ^ can be defined exactly i n the energy equation but i s an approximation when used 1 r> to supply the speed of sound , or information about the microscopic structure (degrees of freedom) of the gas. Having outlined the methods of s o l u t i o n of the stationary shock jump equations, we now evaluate the Influence of l o c a l v a r i a t i o n s of the medium and external sources on the propagation of the shock -24-CHAPTER II SHOCK PROPAGATION INTO INHOMOGENEOUS MEDIA In t h i s chapter, the influence of l o c a l v a r i a -tions i n the unshoeked f l u i d parameters and i n the external sources on the front v e l o c i t y of an established shock i s examinedo I t i s important to r e a l i z e the e f f e c t s of a l l disturbances i n order to deduce the changes i n par-t i c l e v e l o c i t y ahead of the shock using the SWING probe. Shock propagation into inhomogeneous media i s discussed i n a quite general manner so that i n the next chapter we can pick out those features which makes a shock useful as a flow v e l o c i t y probe. The propagation of shock waves into homogeneous gases i s well understood, but the more general question of how shocks propagate into inhomogeneous madia has not yet been completely answered. In order to consider a l l the possible influences upon a shock, we write the front v e l o c i t y i n i t s f u n c t i o n a l form (general shock front equation) v . v ,*.,<*, - A p , w , s, ft ( ? ) Our aim i s to prescribe the change i n the front v e l o c i t y due to changes (inhomogeneitles)in the various parameters, -A, » ?. • and u, are the equilibrium ( f l u i d dynamic) properties of the gas ahead of the shock. 0 » F » and V7 are the three external sources which act l o c a l l y at the f r o n t . Q. and 9- are the adiabatic exponents - 2 5 -before and behind the front and which describe some of the nonequilibrium thermodynamic properties of the gas. 3 X i s the s p a t i a l configuration (one, two, or three dimensional flow). ^ Is the influence of the d r i v i n g mechanism upon the front v e l o c i t y or i n other words, the influence of external sources which act some distance behind the fronfcv The study of the front v e l o c i t y dependence upon these variables i s a very d i f f i c u l t problem and one which deserves some s i m p l i f i c a t i o n r i g h t from the outset. Henceforth we s h a l l ignore the influence of mass sources since D i s usually n e g l i g i b l y small (eg as i n the important case of fusion detonations 1?). Also the discus-sion w i l l be r e s t r i c t e d to one-dimensional flow with a constant d r i v i n g mechanism which conforms to the usual types of shocks found i n the laboratory. The remaining variables must be studied i n two d i s t i n c t and separate cases according to whether the inhomogeneous variables are continuous or discontinuous, II 1) SHOCK INTERACTION WITH A DISCONTINUOUS INHOMOGENEOUS REGION Consider a general inhomogeneous region i n which some or a l l of the i n i t i a l conditions vary d i s -continuously. For such a region, there are only two p o s s i b i l i t i e s J a) the pressure i s continuous across the dis c o n t i n u i t y , or b) the pressure jumps across the d i s --26-contlnulty. I f the pressure Is continuous, then the d i s -continuity Is of the form of a contact surface separating two d i f f e r e n t gases or ene gas at two d i f f e r e n t temper-atures. Since d i f f u s i o n e f f e c t s must always occur at contact surfaces, t h i s class of d i s c o n t i n u i t y may not be t r u l y discontinuous. Thus contact surfaces are treated as continuous inhomogeneous media and the s o l u t i o n holds even i f a contact surface were i d e a l l y discontinuous^ 8 . Therefore we are only l e f t with the case where the pressure changes discontinuously. Of course the same general conservation equations must hold across the inhomogeneous d i s c o n t i n u i t y as held across a shock wave (ie the Rankine-Hugoniot equations Including source terms) 0 I t i s convenient to consider an e f f e c t i v e energy source term which covers the e f f e c t s o f external forces as well as energy sources. The conservation equations f o r the general one-dimensional pressure d i s c o n t i n u i t y become ? ^ * < ? j A r ( i Q ) %-> t t * " 3,-1 ?, where \>J* is the e f f e c t i v e energy input. These equations have been solved exactly and the solutions a r e ^ i - 2 7 -( 1 3 ) . (1*0 (15) where - i f and € is small f o r both large and small Mach numbers. The Ranklne-Hugoniot equations f o r an adiabatic shock represent the s p e c i a l form of ( 13)-(15) when \+] s Q ( p o s i t i v e sign i n equations 13-15) . These jump equations quite generally describe the change i n pressure, density, and p a r t i c l e v e l o c i t y across a pressure d i s c o n t i n u i t y where and (16) (17) Thus we consider the effects of the external source terms i * by considering only one parameter w or more conveniently ft which i s defined asi / ± -a e g * - A W*. (18) As the fronts are discussed here, each s p e c i f i c physical phenomena i s considered as a r e l a t i o n s h i p be-tween the external force f i e l d F and the energy source W . Therefore i n d i v i d u a l examples are expressed as - 2 8 -a r e l a t i o n s h i p between F and W to uniquely define \J (and thus ^ ). In t h i s sense, [5 is a parameter describing the flow which allows us to ignore the actual physics of the external sources i n order to concentrate on the r e s u l t i n g f l u i d behaviour. We want to consider what happens when two such generalized waves c o l l i d e . The general i n t e r a c t i o n scheme appears i n figure 6, We l a b e l the various equilibrium regions by numbers and denote the interface between two such regions 1 and j as • Using t h i s notationj ^1,2) i s the incident shock t r a v e l l i n g with v e l o c i t y V , into the cold gas and having a known external source parameter A3., . <i.5> i s the incident d i s c o n t i n -u i t y and Is t r a v e l l i n g with v e l o c i t y into the c o l d gas. The equations ( 13)-(15) hold across (l*5) with the known external source parameter . & t j ) and<^,5> are also generalized waves t r a v e l l i n g into the s i n g l y shock heated gas with v e l o c i t i e s V 23 and Nj^. r e s p e c t i v e l y . Again equations ( 13)-(15) hold across these d i s c o n t i n u i t i e s with known parameters and A/^ • ^ 3 i ^ i s a contact surface separating the two doubly shock heated regions. Across the contact surface, we have two equations i U % « UH - +f ( 1 9 ) , ( 2 0 ) The pressure and p a r t i c l e v e l o c i t y are continuous across contact surfaces, and i n t h i s case, the contact surface - 2 9 --30-v e l o c i t y equals the p a r t i c l e v e l o c i t y i n regions 3 and Using only the conservation of mass and momentum, we obtaini 1 1 1 (21) Equation (21) i s the basic equation describing the general i n t e r a c t i o n scheme. To complete the s o l u t i o n , we replace the p a r t i c l e v e l o c i t i e s by functions (making use of the energy equation) of the front v e l o c i t i e s and the various *s ( l i k e equation (22)); then equations (21)and (19) are solved i t e r a t i v e l y f o r the v e l o c i t i e s N/^ and X/^ \/«js i s the v a r i a b l e of i n t e r e s t since the wave ^1,2^ has become the wave ^^, 5 ^ and we have found the front v e l o c i t y a f t e r i t has encountered the discontinuous inhomogeneous media, 3?he basic i n t e r a c t i o n scheme (figure 6) i s based upon the well known head-on c o l l i s i o n of two 3 adiabatic shocks , When an adiabatic shock encounters another shock; then shocks are r e f l e c t e d back through the heated gas, A contact surface i s formed which separates these two doubly heated regions. This i n t e r a c t i o n scheme i s the basis f o r many studies of shocks crossing both 9 20-25 continuous and discontinuous inhomogeneous regions 9 . 26 Glass and Patterson have experimentally v e r i f i e d the i n t e r a c t i o n scheme f o r the i n t e r e s t i n g cases of a shock - 3 1 -c o l l l d i n g with a shock? a shock c o l l i d i n g with a contact surface? and a shock c o l l i d i n g with a r a r e f a c t i o n wave. The generalized i n t e r a c t i o n schemes to cover these s i t u a -tions predicted f i n a l wave strengths which were i n good agreement with experiment. Other s p e c i f i c generalizations of the basic shock-shock i n t e r s e c t i o n have been used 27 and we note that c e r t a i n cases w i l l occur i n which the shock i s changed into a r a r e f a c t i o n wave due to i t s i n t e r a c t i o n with the inhomogeneous region. Therefore, we I m p l i c i t l y assume that our shocks remain shocks or at l e a s t that we w i l l be able to recognize when a shock changes into a r a r e f a c t i o n wave. However we should emphasize that we are a p r i o r i assuming that such interactions a c t u a l l y do describe the c o l l i s i o n s of two generalized shocks Just as i t does fo r the c o l l i s i o n s of two adiabatic shocks. Some evidence does exi s t which tends to support the i n t e r a c t i o n model* our smear picture i n figure 20 i s an example of a general-ized i n t e r a c t i o n which i s described by the i n t e r a c t i o n 28 model, as well Rehm obtained this i n t e r a c t i o n model using a perturbation method to describe the absorption of l a s e r r a d i a t i o n at a shock fr o n t . However furture work must involve experimental v e r i f i c a t i o n of the i n t e r a c t i o n model ( f i g . 6) In a wide range of experimental s i t u a t i o n s . Above we have considered a generalized form i - 3 2 -of the in t e r a c t i o n scheme to study a shock crossing a discontinuous inhomogeneous region. Next we consider a d i f f e r e n t generalization to understand the propagation of a shock through a continuous Inhomogeneous region. 9 Following the method of C h i s n e l l , we l e t the inhomogeneity go to the l i m i t of an l n f i n i t e s s i m a l element and consider the i n t e r a c t i o n of the shock with the l n f i n i t e s s i m a l d i s c o n t i n u i t y . II 2) SHOCK INTERACTION WITH A CONTINUOUS INHOMOGENEOUS wave through a one-dimensional 9 continuous inhomogeneous medium where the i n i t i a l parameters each vary smoothly as a function of p o s i t i o n . Equations ( 13)-(15) are now rewritten i n the more convenient formi REG  We now consider the propagation of a shock (22) (23) (24) where IX ^2, and - 3 3 -d i f f e r e n t l a t i n g equation ( 2 2 ) y i e l d s t " /vr, 2 % 7 ^ ( 2 5 ) We now consider the general i n t e r a c t i o n scheme ( f i g u r e 6 ) to c a l c u l a t e d& i n terms of O^, » ^ % * and dfe . In t h i s case ^ 1 . 2 ^ i s the incident shock having a known external source parameter f ^ 2 , 3 ^ i s a re-f l e c t e d wave across which we assume the Rankine-Hugoniot equations hold ( fi^2 2 )• ^ 3 » ^ i s a contact sur-face; ^ » 5 ^ i s the refra c t e d shock having an external source parameter a n d a strength i ^ j * ^ i l * ^ ' and ^ 1 » 5 ^ is an i n f i n i t e s s l m a l element of the inhomo-geneous region across which us+u +dut ; ? 5 s ? ^ d ? ( \ jQs + dfit ( 2 6 ) Using the two equations ( 1 9 ) and ( 2 0 ) across the contact surfaceJ implies that d^9, Of ^  while implies that M 1 -s (28) - 3 4 -Substituting equations (24) and (27) i n t o (28) and taking to the f i r s t order of smallness y i e l d s i where s u b s t i t u t i n g (29) into (25) y i e l d s t ^ U j ^ H^J^^l^) S> + ^ » / j$ ( 3 0 ) where the c o e f f i c i e n t s are dimensionless and are defined by equation ( 3 0 ) ( 3 D u * > s * " . i s s C r V - ^ ) <32) Pigues 7-10 show the values of the p a r t i a l derivatives (equations (3l)-(34)) as a function of the shock strength 2- and figur e 11 indicates these values f o r a strong p r e s s u r e r a t i o z - 3 6 -p r e s s u r e r a t i o (z) - 3 7 -- 3 9 -N o t i c e t h a t the c o e f f i c l e n t s C when p> = 2.^ o r when VV - < ( i e when the energy l o s t f rom the from the f r o n t e q u a l s the energy swept up by the f r o n t ) . -40-shock as a function of ^ . Equation ( 3 0 ) governs the complete i n t e r a c t i o n of the continuous flow with a shock. Equation ( 3 0 ) i s e s s e n t i a l l y the f i r s t d e r i v a t i v e of equation ( 9 ) with a l l the variables of interest included. Cjz i s not included since the physical meaning of such a term x^ould be d i f f i c u l t to in t e r p r e t and impossible to use. ^ is not included since i t does not appear i n equations (22) to (24) which implies that! r e 1 H O . However t h i s i s true only i f C* ^ which i s v a l i d only at low temperatures (ie when the enthalpy and speed of sound d e f i n i t i o n s of X are i d e n t i c a l ) . I f instead, we re-arrange the variables i n order to avoid using the approx-imation C = ^ ; then a value f o r ^ J can be obtainedt \ * h +. T^tT ( 3 5 ) where (assuming the shock is adiabatic ie ^>~2.0 ) 0 This expression describes the change i n shock v e l o c i t y due to some nonequilibrium behaviour of the gas i n front 14 of the shock. I f f o r instance , we assume that the ions, electrons, and neutrals each have a Maxwellian v e l o c i t y d i s t r i b u t i o n , so that we may a t t r i b u t e temperatures -41-TJ • Te t TQ (which may d i f f e r ) to each of the species, then O r 5 * — — I ( 3 6 ) Equation ( 3 6 ) thus defines g i n a non-equilibrium medium and l o c a l changes i n g would cause changes In a shock's v e l o c i t y propagating through the medium according to equation ( 3 5 ) . In p r a c t i c e , and Cf are d i f -f i c u l t quantities to measure and d e t a i l e d enformation about Ojt (ahead of an actual shock) i s hard to come by. The s p e c i f i c part of the shock front equaton which rel a t e s to the external source parameter (equation (3*0) also describes the absorption of energy at a shook 28 front due to la s e r heating. Rehm has considered the problem of la s e r r a d i a t i o n -Interacting with a shock wave head-on and has found (using a perturbation method) that a secondary shock propagates back through the shock heated gas. In contrast, we apply the generalized C h i s n e l l analysis (figure 12) to obtain the r e l a t i o n between the shock v e l o c i t y and the energy absorbed. Figure 13 compares our r e s u l t s to Renin's, although the res u l t s are not i d e n t i c a l , they do show s i m i l a r behaviour and indicate that an extremely large amount of energy would have to be added to the shock front to r e s u l t i n any change i n the front v e l o c i t y f o r even moderate shock -42-F i g . 12 I n t e r a c t i o n Scheme f o r the A b s o r p t i o n o f R a d i a t i o n a t a Shock t i m e secondary shock contact surface Incident laser beam distance -43-strengths. Both our re s u l t s here and Rehm's r e s u l t s can be written i n the form / 3Vc \ where i s the i n t e n s i t y of the absorbed l a s e r radia-t i o n ; "Xc3 i s a s c a l e i n t e n s i t y equal to the i n t e n s i t y required to s o l e l y support a front as a Chapman-Jouguet detonation with v e l o c i t y V ; and and (^^) a r e dimensionless co e f f e c i e n t s . Both Rehm's and our values f o r L ^ j ; and ^  Q X / have been plotted i n figures 13 and 14. The s l i g h t d i f -ferences i n the values are not as important as the fac t that J. c 3- 0 1 tf( which implies that an extremely large amount of energy i s required to produce even mod-erate changes i n the shock v e l o c i t y as well as moderate secondary shocks. Thus the generalized C h i s n e l l method -presents a simple alternate d e r i v a t i o n which complements Rehm's o r i g i n a l r e s u l t s . Note that T - AJjW and that we are considering the case when F- O and W - W • The above discussion on shock propagation into continuous inhomogeneous regions has a quite general formulation but a l l questions have not been completely answered. The general problem had been solved p a r t i a l l y 9 i n several previous studies. C h l s n e l l considered shock propagation into a region of continuously varying density but with constant i n i t i a l pressure. Ono, Sakashlta, and Yamazaki 2^ extended Ch i s n e l l ' s r e s u l t s to include pressure v a r i a t i o n as well as density v a r i a t i o n , but t h e i r r e s u l t s could not be written a n a l y t i c a l l y . Whitham^0 presented a more general study, and was able to obtain a n a l y t i c r e s u l t s f o r shock propagation into a region of continuously varying density and pressure. Whitham's res u l t s are not completely general, since he made several assumptions about the nonuniform region. He used the c h a r a c t e r i s t i c F i g . 13 D i m e n s i o n l e s s C o e f f i c i e n t ^ Q-7" 5 Z present results Rehm's resul ts 10 Nbi A l t h o u g h the d i f f e r e n t t h e o r e t i c a l approaches y i e l d q u i t e d i f f e r e n t v a l u e s f o r the c o e f f i c i e n t s ; the two approaches y i e l d s i m i l a r v a l u e s f o r the secondary shock v e l o c i t y and the i n c i d e n t shock v e l o c i t y ( i . e . the a c t u a l p h y s i c a l e f f e c t s ) s i n c e the term i n which the c o e f f i c i e n t s appear i s s m a l l ( e q u a t i o n ( 3 7 ) ) . ,3 F i g . 14 D i m e n s i o n l e s s C o e f f l c i e n t V <- " ' J . 2 J 3 1 ; Rehm's results for ''short times present results, - 4 6 -equation, and assumed that the pressure and density-d i s t r i b u t i o n s are maintained i n equilibrium by a force f i e l d or momentum source . However, i n general, the non-uniform region may f i r s t be unsteady, and secondly have energy and mass sources appearing i n the equations of motion, Whitham also makes the strong shock approximation so as to ignore the force term i n the c h a r a c t e r i s t i c equation. In order to avoid these assumptions, a general study should consider the three f l u i d dynamic quantities as completely independent parameters as we have done. and research needs s t i l l to be done on the general question of shock propagation into inhomogeneous media. S p e c i f i c a l l y , the various generalizations of the i n t e r -a ction scheme need to be j u s t i f i e d experimentally. As well r e l a t i o n s between the various i n i t i a l parameters could perhaps be u n i f i e d i n order to present a simpler approach. The d e r i v a t i o n of equation ( 3 0 ) has been greatly aided by discussions with Prof. R.F. C h i s n e l l . Prof. and has convinced us that his d e r i v a t i o n (equation ( 3 1 ) ) i s correct. Therefore as an aside, we consider another However equation ( 3 0 ) i s not the complete answer C h i s n e l l questioned our de r i v a t i o n of i n Appendix B. - 4 7 -II 3 ) EXPERIMENTAL APPLICATION OF THE SHOCK FRONT EQUATION Equation ( 3 0 ) has general a p p l i c a b i l i t y since no assumptions are made about s p e c i f i c r e l a t i o n s h i p s amongst the downstream v a r i a b l e s . Of course, i n many experiments such r e l a t i o s h i p s w i l l e x i s t and can be used. As a simple a p p l i c a t i o n of the shock front equation we have used i t to predict the l o c a l v a r i a t i o n i n shock front v e l o c i t y through a known r a r e f a c t i o n wave. In order to generate both the r a r e f a c t i o n wave and the shock wave we have spark-ignited a plane l i n e a r detonation i n equi-molar acetylene-oxygen mixtures at an i n i t i a l pressure of 200 Torr. ( f i g . 16) (The apparatus is described more f u l l y i n Appendix A). The Chapman-Jouguet detonation proceeds at a constant v e l o c i t y of 2 , 8 Km/sec and i s immediately followed by a centered r a r e f a c t i o n wave. The detonation i s incident upon a l u c i t e r e f l e c t o r , and a shock wave i s r e f l e c t e d back through the r a r e f a c t i o n wave. ™ — GAS 7 DETONATION VELOCITY (km / . ' secJ INITIAL SHOCK VELOCITY (km/ . . ' sec J 25% He 1.32 2.84 .95 15% Ar 1.27 2.36 .85 33% Ar 1.3G 2.26 .78 50% Ar 1.44 2.15 .76 I I 1 F i g . 16 Shock P r o p a g a t i o n Through a Known R a r e f a c t i o n Wave - 4 9 -This shock should obey equation ( 3 0 ) . We assume that a secondary reaction does not occur i n the r e f l e c t e d shock, because a l l of the oxygen should be used up i n the i n i t i a l detonation. The d i s t r i b u t i o n of the p a r t i c l e v e l o c i t y , density, and pressure through the r a r e f a c t i o n wave have 31 been given by Taylor^ , The p a r t i c l e v e l o c i t y i s l i n e a r with respect to distance and i s zero at the r a r e f a c t i o n t a i l . The density and pressure d i s t r i b u t i o n s are found assuming ^ = constant, and are given byi X * U - 2 F Q ) (38) (39) where Y i s taken to be constant, Y =1.2 . The shock v e l o c i t y was calculated i t e r a t i v e l y as a function of p o s i t i o n , using a simple predictor numerical i n t e g r a t i o n formula f o r equation ( 3 0 ) , and using the measured value of 1 , 15 Km/sec f o r the s t a r t i n g v e l o c i t y Immediately at the r e f l e c t o r . The gas mixture was changed by mixing various amounts of i n e r t gases (argon and helium) into the equimolar acetylene-oxygen. The detonations now proceed at a d i f f e r e n t v e l o c i t y , which implies that a d i f f e r e n t , but s t i l l known rare-f a c t i o n wave was created. The i n i t i a l pressure was increased to 300 Torr to ensure that a detonation would - 5 0 -s t i l l be produced. The values f o r the detonation v e l o c i t y , i n i t i a l r e f l e c t e d shock v e l o c i t y , and the adiabatic constant appear i n the table. The experimental values appear i n figures 1? a n d l 8 . The experimental points tend to agree with the t h e o r e t i c a l predictions (equation ( 3 0 ) ) although the agreement is not exact. A c t u a l l y there are several d i f f i c u l t i e s inherent In even as simple as experiment as th i s one. I t i s e a s i l y noticed i n the o r i g i n a l negatives of the smear pictures ( f i g . 16) that a weak disturbance interse c t s the r e f l e c t e d shock at about 3 cm from the r e f l e c t o r . This disturbance i s generated at the i g n i t i o n end of the tube. Ma i n t e r p r e t t h i s disturbance to be a weak compression wave which changes the pressure, density, and p a r t i c l e v e l o c i t y d i s t r i b u t i o n s (equations ( 3 8 ) and ( 3 9 ) ) through the r a r e f a c t i o n wave. We have assumed that t h i s weak disturbance only a f f e c t s the ra r e f a c t i o n wave a f t e r the disturbance. However we have no evidence to support t h i s supposition. Obviously any r e a l l y c a r e f u l experiments must involve a cleaner t r i g g e r i n g of the discharge (perhaps by a l a s e r spark) and c a r e f u l design of the detonation chamber to reduce r e f l e c t i o n s (perhaps using a spherical chamber, although equation ( 3 0 ) would have to be put into three dimensional form). Another d i f f i c u l t y i s that the Mach numbers of the r e f l e c t e d shocks are very small (of the order of 1.6). For weak shocks the influence of doubly r e f l e c t e d - 5 1 -waves must be c o n s i d e r e d ^ ' C h i s n e l l ^ had also con-sidered the e f f e c t of the wave r e f l e c t e d by the v a r i a b l e medium when the s i n g l y r e f l e c t e d wave passes back through i t . The doubly r e f l e c t e d wave moves i n the same d i r e c t i o n as the incident shock. He found good agreement fo r .the s i n g l e r e f l e c t i o n theory whenever the r e f l e c t e d wave was small (ie whenever the incident shock was strong or the density gradients were not too strong). However, we have no c r i t e r i a f o r deciding when these factors become important, although i n the strong shock approximation ( W ^ 5 ) , doubly r e f l e c t e d waves do not 1 B appreciably influence the r e s u l t s . A t h i r d d i f f i c u l t y i s the method of measuring the front v e l o c i t y . The smear camera i s a c t u a l l y recording only luminous fronts and therefore luminous e f f e c t s can perturb the i d e n t i f i c a t i o n of the luminous front with the shock f r o n t . The most pronounced e f f e c t Is the flooding of the photographic f i l m (both polaroid and Trl-X) i n the region of the r e f l e c t o r ( f i r s t centimeter). The e f f e c t can be seen as the instantaneous width of the shock-heated region when r e f l e c t i o n occurs and changes the measured luminous front v e l o c i t i e s f o r the f i r s t centimeter from the actual shock v e l o c i t i e s i n that region. I f the curves i n figures 17 and 18 were normalized with the experimental values at the 3 cm mark Instead of at the r e f l e c t o r , a l l the curves would f i t the experiment - 5 3 -F i g . 18 Shock V e l o c i t y Through S e v e r a l Known R a r e f a c t i o n Waves 1.8 "~ ~~ " - 5 4 -f o r distances greater than one centimeter. A fourth d i f f i c u l t y i s spurious effects at the r e f l e c t o r due to the i n t e r a c t i o n of the hot gas with the l u c i t e r e f l e c t o r . Bits of l u c i t e may be bo i l e d o f f the wall and may perturb the r e f l e c t e d shock while i t i s close to the r e f l e c t o r . As well the cooling of the gas by the r e f l e c t o r may be a source of r a r e f a c t i o n waves which would eventually catch up to and slow the r e f l e c t e d shock. The experiments do tend to v e r i f y the shock front equation ( 3 0 ) although we emphasize that more work i s needed. We have, as the next step, inverted the analysis i n order to r e l a t e the shock propagation uniquely to the properties of the unshocked gas. This study l e d to a new approach f o r measuring properties i n an unknown but reproducible plasma flow; as w i l l be seen i n the next chapter. -55-CHAPTER III THEORY OF THE SWING 6PROBE The standard methods of measuring f l u i d dynamic parameters inside an unknown flow f i e l d may be divided into two groups. The f i r s t group comprises methods i n which waves or signals are analyzed which are n a t u r a l l y emitted by the flow f i e l d . Examples of th i s group are the measurement of temperature, density, and p a r t i c l e v e l o c i t y from the t o t a l i n t e n s i t y , shape, and Dpppler s h i f t of s p e c t r a l l i n e s . These measurements do not disturb the flow f i e l d at a l l . The second group consists of probes or external f i e l d s which are applied to the unknown flow. P i e z o e l e c t r i c pressure transducers, magnetic probes, magnetic f i e l d s f o r vxB measurements, tracer methods, and electron beam methods are examples of the second group 1 0. In these methods, a perturbation Is forced upon the flow f i e l d . The measured data therefore has to be corrected or one has to make the perturbation i n f i n i t e s -simally small so that the measured parameters are suf-f i c i e n t l y representative of the undisturbed flow. From the.theory of shock propagation into inhomogeneous media, one can derive a probing technique which goes to the other extreme and uses a large amplitude perturbation - a shock wave- as the diagnostic t o o l . As shown i n Chapter I I , a shock responds sensibly to va r i a t i o n s i n the unshocked gas downstream and mathematical r e l a t i o n s f o r the shock v e l o c i t y as a function of the - 5 6 -dovmstream parameters are a v a i l a b l e . Also the shock wave tr a v e l s with supersonic v e l o c i t y - f a s t e r than small scale perturbations can "warn" the flow downstream. The shock therefore always encounters the flow f i e l d i n the undisturbed condition. Since the probing technique i s based on the analysis of Shock Maves In Non-uniform 33 Gases, we c a l l i t the SWING probing technique.-' Now we turn to the information we need to use a shock wave to probe a general flow f i e l d . A flow f i e l d i s completely described by i t s f l u i d dynamic propertiesJ density, pressure, and p a r t i c l e v e l o c i t y . So we need to know how the shock v e l o c i t y changes as a function of these i n i t i a l changing f l u i d parameters. If we consider a " t y p i c a l " unknown flows the inhomogeneous region w i l l be bounded by a surface of di s c o n t i n u i t y where the f l u i d dynamic parameters have changed abruptly while the i n t e r i o r of the region w i l l feature smoothly varying parameters. This implies that i n order to use the shock as a t o o l KNOW to probe a flow f i e l d , we must Awhat w i l l happen when the shock crosses both discontinuous and continuous regions of the f l u i d parameters. Due to t h i s basic nature of the unknown flow, we must consider the SWING probe i n two e n t i r e l y separate s i t u a t i o n s . - 5 7 -III 1) PROBING DISCONTINUOUS UNKNOWN PLOWS We now use equation (21) which Is the basic equation describing the i n t e r a c t i o n scheme (fig„ 6). To complete the so l u t i o n , we use equation ( 19) and solve f o r LAf i Equation (40) represents the complete s o l u t i o n because once one has , one also knowas As* » -f^s* »and f^-from the conservation equations ( 1 3 - 1 5 ) * Therefore a l l the parameters immediately behind the edge of a discon-t i n u i t y can be found from equation (40) using the SWING probe which only requires that one i s able to measure» Ideal l y then, one makes an adiabatic shock encounter the unknown front with the c o l l i s i o n producing an i n t e r -a ction scheme l i k e that i n f i g u r e 6. Measurement of the relevant front v e l o c i t i e s y i e l d s the p a r t i c l e v e l o c i t y (equation (40)) and thus a c l a s s i f i c a t i o n of the front ^ 1 , 5 ^ i n terms of the external source parameter and the amount of e f f e c t i v e energy being absorbed at the fr o n t . The c l a s s i f i c a t i o n i n the table below i s v a l i d only I f p s Q ( W » W ) so that the e f f e c t i v e energy input i s the actual energy enput. However the parameters behind the front are the same f o r each value of - 5 8 -of (equations 13-15) CLASSIFICATION OF <1.5> luminous shock - 2.o adiabatic shock / < fts < t overcompressed detonation - /.o Chapman-Jouguet detonation < (.0 breakdown wave Thus the SWING probe yie l d s Information about the equilibrium conditions immediately behind the edge of the t y p i c a l unknown flow f i e l d . However, we no longer have an adiabatic shock propagating into the flow i n t e r i o r . Rather the probing wave ^ 4 , 5 ^ w i l l , i n general, have ah e f f e c t i v e energy input appearing at the f r o n t . £ or ) can be found using the conservation equations since one knows the p a r t i c l e v e l o c i t y - V^y behind the front ^ 4 , 5 ^ . I l l 2) PROBING CONTINUOUS UNKNOWN FLOWS Equation ( 3 0 ) i s e s s e n t i a l l y the basic equation describing shock propagation into continuous inhomogeneous media and governs the complete i n t e r a c t i o n of the un-known flow f i e l d with a shockiwave. For a general flow, the four parameters U , , ^ , and are a l l independent; therefore, i n the general case, four measure-ments of the change i n the shock v e l o c i t y would be required -59-to solve equation (30) and thus determine the change i n the f l u i d parameters. These four measurements would be based upon the i n t e r a c t i o n of four shocks (each having a d i f f e r e n t strength) with the unknown flow f i e l d . This severely l i m i t s the general usefulness of the technique as i t places a need f o r quite r i g i d r e p r o d u c i b i l i t y on the unknown flow f i e l d . B y t h i s we mean that the flow f i e l d w i l l have to be the same on four d i f f e r e n t shots i n order that we may solve the four l i n e a r equations with any meaning. The s i t u a t i o n becomes greatly s i m p l i f i e d when the i n i t i a l shock i s a strong one ( i e ^ / ), In t h i s case ( — \ —^ o ( f i g . 11) so that no matter how large:the pressure v a r i a t i o n , the shock v e l o c i t y w i l l not change* Also the scale i n t e n s i t y becomes 9 AT ^ "T--.—^ -1—-i—- v , therefore unless the i n t e n s i t y change i n the e f f e c t i v e energy Input i s extremely large, we w i l l havet (Recall that t h i s was the main conclusion reached by both Rehm's work and our work about the absorption of l a s e r r a d i a t i o n at shock fronts i n Chapter I I ) . Therefore f o r strong shocks, we are able to ignore effects due to inhomogeneitles i n the pressure and i n the external sources, and equation (30) becomesi d v = ^ s u ( ) d u , + ^ A s v % ( M ) where - V * ^ , -60-This Implies that i f we probe the flow f i e l d with several (at l e a s t two) shocks each having a d i f f e r e n t we should obtain a st r a i g h t l i n e . The l i n e w i l l have changes i n density and p a r t i c l e v e l o c i t y w i l l be found throughout the continuous region ( i n t e r i o r of the t y p i c a l unknown flow), s i m p l i f i c a t i o n which can r e s u l t from a few p r a c t i c a l assumptions about the behaviour iaf the unknown flow (for use i n Chapter IV). Consider the propagation of a shock through a continuous region which i s s i m i l a r to a r a r e f a c t i o n wave. This i s a f a m i l i a r occurrence covering the expansion of a gas under many d i f f e r e n t i n i t i a l and boundary conditions. For such a case we have the following assumptions! v e l o c i t y and plot the measured changes AV against ' ATJ Thus the We also examine another example of the a ) Z since external source terms are absent; since -p^ = constant,and i f we take decreasing and U increasing. The equation governing the shock propagation through the unknown region reduces to; oiV - is* du (42) - 6 1 -where ( 4 3 ) Using equation ( 4 2 ) , a measurement of the change of the shock v e l o c i t y throughout the inhomogeneous region y i e l d s a value f o r the changing p a r t i c l e v e l o c i t y i n that region, Sample values f o r < s * are shown i n figu r e 19 while f o r strong shocks is given by equation ( 4 4 ) assuming III 3 ) PROPERTIES OP THE SWING PROBE The advantages of the SWING probe may be summarized as followsJ J a) I t does not perturb the unknown flow u n t i l a f t e r the measurement i s completed, i f we can assume that there i s no coupling between the various regions as f o r example by precursors, r a d i a t i o n , or magnetic f i e l d s . In f a c t , coupling from the flow f i e l d a f f e c t i n g the of the probing shock Is i n p r i n c i p l e unimportant since we now probe with a shock having a new value, however coupling from the shock to the unknown flow w i l l perturb the unknown flow before the shock encounters i t . b) The method r e s u l t s i n a determination of the p a r t i c l e v e l o c i t y , density, pressure, and l o c a l xx (assuming i s measureable) -62-F i g . 19 Approx imate Shock C o e f f i c i e n t <& 5 10 p r e s s u r e r a t i o (z) - 6 3 -energy input at the edge of the flow as long as the generalized i n t e r a c t i o n scheme ( f i g . 6) may be assumed to apply. A region i n the unknown flow i s considered continuous or discontinuous according to our a b i l i t y to measure distances (eg i n f i g u r e 26, the properties at the leading edge of the flow are c l e a r l y discontinuous on the scale of the smear pictures while the properties between the leading edge and the luminous front are c l e a r l y continuously varying on t h i s same s c a l e ) . S i m i l a r l y discontinuous and continuous probing si t u a t i o n s w i l l always be defined by the a b i l i t y to measure distances i n the flow. The minimum possible scale distance In a flow i s determined by the shock thickness (that i s our measuring device must always consider the shock front to be discontinuous as this i s the basic assumption defi n i n g the a p p l i c a t i o n of the conservation equations). Usually the molecular shock thickness i s of the order of 10 mean free paths i n the hot gas (eg 5 x 10 mm i n 1 Torr argon f o r an adiabatic shock with M= 8). Occasionally the t o t a l shock thickness w i l l be very-large due to the r e l a x a t i o n of i o n i z a t i o n . In these cases the probing shock should be described by the properties immediately behind the front since these properties ( ft and $x ) describe that actual propagation of the front through the unknown flow. - 6 4 -Considerations of shock thickness thus r e s t r i c t the SWING probe to applications at f a i r l y high ambient pressures where shocks may be more e a s i l y considered • as discontinuous. c) The p a r t i c l e v e l o c i t y and density d i s t r i b u -tions can be obtained throughout the continuous region using strong shocks. The p a r t i c l e v e l o c i t y i s usually d i f f i c u l t to obtain by other methods. However a wide range of strong shocks are usually d i f f i c u l t to generate and as well the Mach number of the probing shock i s usually severely reduced upon c o l l i s i o n with the edge of the unknown flow. The SWING probe represents a general improvement i n the entir e spectrum of flow v e l o c i t y measurements because i t has a wide range of a p p l i c a t i o n . The technique works best using strong probing shocks (f o r s i m p l i c i t y i n the equations). However the measuring system should be optimized f o r detecting changes i n the. shock v e l o c i t y which are of the order of the changes i n flow v e l o c i t y . d) The r e s u l t can be simply coupled to other information about the unknown flow. For instance, i t i s quite easy to determine the pressure using p i e z o e l e c t r i c probes and t h i s information can be e a s i l y f i t t e d into the general shock i n t e r a c t i o n scheme to produce more r e l i a b l e information about the other parameters. As well there Is also the p o s s i b i l i t y that the entire method may be treated more e f f e c t i v e l y by measurements of the - 6 5 -pressures i n the various regions instead of the front v e l o c i t i e s , e) Many usefu l s i t u a t i o n s w i l l often a r i s e which w i l l r e s u l t i n great s i m p l i f i c a t i o n s of the neces-sary equations. For instance, i n many cases ft w i l l be constant ( fl "S implies that we are probing with an adiabatic shock everywhere while / - 0 implies that the probing device i s a Chapman-Jouguet type wave). Often simple r e l a t i o n s w i l l e x i s t between the variables i n the unknown regions which w i l l r e s u l t i n further s i m p l i f i c a t i o n . For instance, -f° ^  = constant may be assumed. - 6 6 -CHAPTER IV PARTICLE VELOCITY MEASUREMENTS OP A T-TUBE FLOW USING THE SWING PROBE One of the fundamental properties of a dense flow f i e l d i s the l o c a l average v e l o c i t y . Numerous ap-34-proaches are taken to obtain t h i s quantity-; however accuracies even under i d e a l conditions are usually of the order of & 1 Km/sec35-38 and often l o c a l perturbations of the flow are required to perform the measurement. The SWING probe i s capable of measuring flow v e l o c i t i e s to the order of + .1 Km/sec and the flow i s perturbed only a f t e r the measurement has been taken. As an a p p l i c a t i o n of the SWING probe, we •49 used i t to probe the flow f i e l d produced by a T-tube. 8 The review a r t i c l e by Muntenbruch has summarized many years of controversy regarding the flow f i e l d produced by a convential electromagnetic shock tube. E s s e n t i a l l y T-tubes produce non stationary shock waves of the bl a s t wave type i f the plasma of the d r i v i n g discharge does not advance ri g h t into the shock front and appreciably influence the plasma formed there. Since we can obtain the p a r t i c l e v e l o c i t y without any assumptions about force and energy sources i n the unknown flow, the flow In a T-tube was probed at ambient pressures of 1 . 5 and 6 . 5 Torr Argon. The measurements on the one hand confirm the accepted d e s c r i p t i o n of the T- -tube flow® but on the other hand also i n v i t e comparisons of the T-tube flow with other step wave phenomena. - 6 7 -The T-tube flow can be divided into three regimes. In the f i r s t regime 9 the shock and luminous front cannot be distinguished from one another. This regime i s characterized by high front v e l o c i t i e s and occurs mainly i n the v i c i n i t y of the d r i v e r section and at low ambient pressures. I t i s not so much the distance from the spark gap but rather the v e l o c i t y of the shock fro n t , and hence of the discharge plasma behind as wellg that defines t h i s v e l o c i t y range. The plasma of the spark discharge mixes with the gas heated In the shock fr o n t . This regime was produced i n th i s experiment f o r a duration of about 80 microseconds at a r e l a t i v e l y low ambient pressure of 1.5 Torr Argon and the SWING probe was used to obtain d e t a i l e d information about the flow parameters. The second regime occurs as the front v e l o c i t y decreases with t r a v e l alowsthe tube. The shock front and the luminous front continue to separate so that a region i s formed i n which the plasma data are mainly determined by the shock wave. This intermediate region i s occasion-a l l y entered by plasma wisps. These plasma transport energy forward from the discharge plasma cloud to the gas behind the shock fr o n t . We were unable to reproduce thi s type of regime. The t h i r d regime occurs as the shock Mach number further decreases and the space behind the shock f r o n t p no longer invaded by these b i t s of plasma keeps -68-F i g . 20 Experimental Setup h- lxl ^ V) 0Q -69-getting bigger and constitutes a region of the gas which i s s o l e l y heated and compressed by the shock wave. The properties of t h i s regime and of shock waves i n p a r t i c u l a r are extremely well known so that the probing of t h i s regime may be considered more of a check on the SWING probing technique rather than a d i r c t a p p l i c a t i o n of the methodi This work i s the f i r s t experimental a p p l i c a t i o n of the technique and thus the v a l i d i t y check i s nne main objective. A higher ambient pressure of 6.5 Torr Argon was used to obtain t h i s low v e l o c i t y regime IV 1) DESCRIPTION OP THE EXPERIMENT The experimental arrangement i s shown i n figure 20,,further construction d e t a i l s of the apparatus appear i n the appendix. The T-tube on the l e f t i n j e c t s a plasma flow f i e l d i n the 25mm diameter test section. The probing 39 shock, generated with a small detonation d r i v e r i s mounted on the r i g h t end of the test section. Each trace of the shock wave through the unknown flow f i e l d y i e l d s plasma parameters along t h i s path of the shock wave i n the space-time diagram recorded by the smear camera. Therefore many d i f f e r e n t photos of encounters must be recorded i n order to scan the-entire flow f i e l d . The smear camera i s used to t r i g g e r the detonation i n the shock d r i v e r . The detonation implodes on axis and bursts -70-the Mylar diaphragm. The detonation products then drive a shock i n the argon test gas which propagates towards the T-tube end. The detonation t r i g g e r spark i s used (through a delay unit) to s t a r t the T-tube discharge which propagates towards the approaching shock wave. The delay times are arranged so that the c o l l i s i o n s occur i n the f i e l d o£ view of the smear camera. The measuring system was a standard smear camera used at a writing speed of 15 microseconds/cm. The accuracy of the front v e l o c i t y measurements i s to the order of 0 . 0 5 km/sec and i s li m i t e d by the width of the front as i t appears on the f i l m . A s l i g h t impurity of 3 $ c 2 ^ 2 w a s u s e d to improve the emission properties of the gas and i t was assumed that t h i s impurity d i d not a f f e c t any of the equilibrium states appreciably. The shock generator has some but not a l l of the desireable properties f o r use i n the SWING technique. The SWING technique requires variable high v e l o c i t y shocks which can be made to a r r i v e at a p o s i t i o n defined to within centimeters and tens of microseconds so that the c o l l i s i o n s are reproducible. The t r i g g e r i n g proper-t i e s of t h i s type of shock d r i v e r are excellent because i t has a constant formation time due to the constant, detonation v e l o c i t y . The diaphragm opening times were defined to within one microsecond and the j i t t e r i n the resultant shock v e l o c i t i e s was less than 0 , 0 5 km/sec. One meter (40 tube diameters) was allowed f o r shock - 7 1 -separation to occus? and the ambient pressure was kept high enough (greater than 1 Torr argon) so that a reasonable shock heated region was always a v a i l a b l e (of the order of 1 cm). The v a r i a t i o n i n the c o l l i s i o n p o s i t i o n due to the j i t t e r i n a r r i v a l of both the probing shock and the T-tube flow was less than 0 . 5 cm f o r the 6 . 5 Torr argon case and less than 3 cm f o r the 1 ,5 Torr argon case. In the high v e l o c i t y regime, the T-t.ube front had a v a r i a t i o n of + 0 . 2 5 km/sec i n i t s v e l o c i t y so that i t s a r r i v a l time was correspondingly more uncertain. Since the ambient pressure had to be kept constant i n order to have a reproducible T-tube flow, v a r i a t i o n s i n the probing shock v e l o c i t y had to be arranged by changing the i n i t i a l pressure of the C2^2"QZ i n the d r i v e r . A range of only 300 to 500 Torr CgR^-Og was a v a i l a b l e since below 300 Torr secondary r e f l e c t i o n s predominate i n the d r i v e r chamber and above 500 Torr the chamber was damaged. This v a r i a t i o n i n d r i v e r pressure allowed a v a r i a t i o n of only one Mach number (from ^i2~ ^ to M 1 2= 7 ) i n the probing shock v e l o c i t y . Unfortunately the a v a i l a b l e d r i v e r chamber r e s t r i c t s the a p p l i c a t i o n of the SWING probe i n continuous regions of the T-tube flow (property c) page 6 4 ) . The duration of the unknown flow f i e l d was extended i n order to allow reproducible c o l l i s i o n s between i t and the shock. This was accomplished by using a lumped delay l i n e to drive the T-tube. A constant current was produced f o r A^"= go f+ sec which established a constant flow region f o r that time. Further d e t a i l s on the apparatus are presented i n the appendix. IV 2) THE HIGH VELOCITY (LOW PRESSURE) REGIME A t y p i c a l smear picture f o r the c o l l i s i o n between the adiabatic shock and the T-tube flew at an ambient pressure of 1 .5 Torr argon appears i n figure 2 0 . A l l the relevant fronts are v i s i b l e and the front v e l o c i t i e s ^IZ * ^22 ' ^li ' * a n d ' Ys1 ( f igure 2 1 * w e t e m e a s u r e d from the smear p i c t u r e s . Using equation (40) the flow v e l o c i t y Uig i s obtained d i r e c t l y (figure 2 2 ) . Equation (40) y i e l d s the a c t u a l flow v e l o c i t y immediately behind the front and the equation i s independent of any force terms or energy source terms which may appear and which complicate the s o l u t i o n of the Rankine-Hugoniot equations so that a simple measure of V^-does not reveal u f A U < • A measurement of r r y i e l d s a value f o r [J.r so that a c l a s s i f i c a t i o n of ^1*5^ i s possible i n terms of the amount of e f f e c t i v e energy being absorbed at the fr o n t . Using equation (40), the parameter fif£ has been found and i s plot t e d i n f i g u r e 2 3 . There i s considerable s c a t t e r i n the actual value f o r ft> but except f o r a few points fi^ I i s found. The T-tube flow thus behaves l i k e a breakdown wave i n t h i s high v e l o c i t y (low i n i t i a l pressure) regime. - 7 3 - . F i g . 2 1 F r o n t V e l o c i t i e s f o r the C o l l i s i o n s i n the Low P r e s s u r e Regime 3-0 o 2-0 © \ E JUL ~ 1-0 >-o 0-0 txJ > H-0 -2-0 -3-0 h v v © v 4 5 A V I 2 • v 3 4 V v 2 3 0 V I 5 D • D q en • • c P • • W 9 v V 0 0 0 o 0 V 0 XT • • DISTANCE (cm) '45 '34 23 15 10 - 7 4 -F i g . 22 Flow V e l o c i t y Immediate ly Beh ind the T-Tube F r o n t a t Low P r e s s u r e s 3 - 0 o to 2 - 0 o 3 U J > 1-0 o V I5 A u 5 CP o o o o° °o°8co0 O O n ° o o o 8 CO 0 ° ° ° o o o o o o oo ° o o o o o A A A A A A A . A A A A ^ A A A A A A^ A A A A A A & A A A io DISTANCE (cm) o 15 - 7 5 -F i g . 23 E x t e r n a l Source Parameter f o r the T-Tube Flow A P = 6-5 torr Ar O P = 1-5 forr Ar ADIABATIC S H O C K /A A AA A^  AAA^S§£-o o 8 o °°oo°° o° 0° 8 8 O O ° °o oo o o° o ° • ° ° §° ° °o ° o ° OVERCOMPRESSED DETONATION C H A P M A N -vIOUGUET DETONATION. 8 OVEREXPANDED DETONATION 1 0 2 0 X (cm) - 7 6 -Since the front <^1,5^ does not change i t s v e l o c i t y appreciably once i t has penetrated the T-tube flow (figure 20), the flow v e l o c i t y U ^ -can be taken as roughly constant throughout the T-tube flow. The large scatter i n the r e s u l t s i s p a r t l y due to the sc a t t e r i n the v e l o c i t y ( i e due to the non r e p r o d u c i b i l i t y of the unknown flow from shot to shot). The analysis using equation (40) Is independent of the energy absorption i n ^ 2 , 3 ^ and ^ 4 , 5 ^ , In p r i n c i p l e these fronts w i l l not be adiabatic shocks but w i l l have e f f e c t i v e energy sources represented by fi^ and ^ y , I f the various possible values f o r » % and are considered as functions of / S j j and (assuming reasonable adiabatic constants) and are compared with the average measured values, then f>*\ - ft is - t .2 (figure 24) which indicates that these waves behave l i k e overcompressed detonations. Using these values f o r , , a n d ^he various adiabatic constants, the v e l o c i t i e s expected from a c o l l i s i o n of <^l ,5^with the adiabatic shock ^ 1 , 2 ^ can be derived and are shown i n figure 21 to be s e l f consistent with the experiment. This i s i n t e r e s t i n g since the experimental c o l l i s i o n s conform to the generalized i n t e r a c t i o n scheme analysis (figure 6 ) , The experimental values f o r can be i n t e r -preted i n terms of various models f o r the T-tube flow -77-F i g . 24 E x t e r n a l Source Parameters f o r the Secondary Waves T 1 1 1 1.0 1.5 2.0 e x t e r n a l s o u r c e parameter ( - 7 8 -f i e l d . As the problem i s posed here, the models are considered as relationships between the external force f i e l d F and the energy source term W. IV 3) MODELS OF THE T-TUBE FLOW a) One possible i n t e r p r e t a t i o n i s to assume that the J. x B force has no actual e f f e c t on the equilibrium properties of the heated gas. Instead the energy source term takes into account the e f f e c t (Joule heating) of the current on the gas while the em d r i v i n g force is li m i t e d to c o n t r o l l i n g the p o s i t i o n of the current and thus of the energy source, This i n t e r p r e t a t i o n amounts to assuming that F=0 and W=W? For this model, the i n t e n s i t y f ^ W ^ d 1 0 1 0 erg/cm 2sec i s calculated from ( 1 8 ) using the measured . The t o t a l energy deposited into the front (W*& t-tube area) i s of the order of 5 joules, while the energy stored i n the capacitor bank i s about 700 joules. The f r a c t i o n of energy delivered into the front i s very small. It could be supplied by the Ohmic heat from a 1 amp current loop passing through the fr o n t , or could be delivered by electron heat t r a n s f e r ^ 0 ' ^ 1 from the main discharge. As a consequence of the v e l o c i t y measurements with the SWING probe, the T-tube flow i n t h i s low pressure regime can be c l a s s i f i e d by the parameter p < I to be a breakdown wave. The equilibrium temperature calculated f o r such a wave i s of the order of 6500°K (calculated from - 7 9 -the equilibrium pressure and density behind the wave). The temperature i n any case must be lower than 8000°K which would be obtained i f the wave of the measured v e l o c i t y was an adiabatic shock,, This low temperature indicates that the equilibrium degree of i o n i z a t i o n at the front i s very low. The pressure calculated f o r such a breakdown wave with the above determined energy source i s t y p i c a l l y 40 Torr or s l i g h t l y less than h a l f the pressure expected f o r an adiabatic shock which i s i n good agreement with Lebedev and Formichev's^ 2 piezo probe measurements. The p a r t i c l e to fs>ont v e l o c i t y r a t i o agrees i n order of magnitude to L i e b i n g ' s ^ q u a l i t a t i v e observations. The c l a s s i f i c a t i o n of <^1,5^ as a breakdown wave Implies that the hot p a r t i c l e s leave the front with a supersonic v e l o c i t y ^ so that i f pressure waves and turbulent elements J were to supply the energy to the front they must t r a v e l at speeds greater than the l o c a l equllbrium speed of sound, b) Another possible i n t e r p r e t a t i o n of ^la5/ is 46 In terms of a transverse i o n i z i n g shock , where the external source terms are r e l a t e d by equation ( 4 5 ) F* ^ O- ? ) (45) V / * € %) (46) where * >/ A)^  i s the Alfven v e l o c i t y . The dimensionless function £ has the l i m i t s -1 f o r the -80-MHD (frozen f i e l d ) case and ~ f o r the ordinary hydro-dynamic case (OHD) where the source terms P=W=0 vanish and the gas i s not ionized at a l l . The experimental value f o r $ = 0 . 7 2 y i e l d s l i m i t s on the r e l a t i o n between F and W. corresponding to the MHD and OHD cases r e s p e c t i v e l y . .$5. Using the experimental value f o r W and solving ( 1 7 ) and (46) f o r the magnetic f i e l d ahead of the wave, a lower l i m i t can be put on Hb( > 350 gauss which cor-responds to the MHD l i m i t . Such a f i e l d may be present. Therefore on the basis of the SWING probe measurement we cannot presently d i s t i n g u i s h whether the wave ^ 1 , 5 ^ i s a breakdown wave propelled s o l e l y by a heat source or a transverse i o n i z i n g shock i n which the magnetic f i e l d supplies an important external f o r c e . However the flow v e l o c i t y immediately behind the front has been measured to a high accuracy and the wave can be interpreted In terms of the .effective energy absorbed as being s i m i l a r to a breakdown wave. IV 4) THE LOW VELOCITY (HIGH PRESSURE) REGIME The SWING probe was also used to analyze the T-tube flow at an ambient pressure of 6 , 5 Torr argon where an adiabatic shock wave i s produced. The purpose of these measurements was to v e r i f y the SWING technique on a well-defined flow. A (distances time) reproduction of the T-tube discharge appears i n figure 25 and indicates probing shock p r o f i l e s as they propagate through the T-tube flow from various c o l l i s i o n p o s i t i o n s . These shock t r a j e c t o r i e s were obtained from about 75 i n d i v i d u a l smear camera photos, of which figures 26a) and 26b) are t y p i c a l examples. A l l the prominent features are evident except the (^1,5^ f r o n t . The <^1,5^ front v e l o c i t y was measured between x = 6 and x = 10 cm from the c o l l i s i o n positions and times. Using t h i s value f o r and measuring V 1 2 i > ^23' v 3 4 » 3 3 1 ( 1 v 4 5 ( f i8« 2 ? ) from the smear pict u r e s , the parameter was again obtained ( f i g . 23). Over th i s distance the front i s ~-c l e a r l y an adiabatic shock wave since =2.0 . This i s expected f o r the T-tube flow i n the low v e l o c i t y regime and the r e s u l t i s the f i r s t v a l i d i t y check on the SWING technique. For distances greater than 10 cm, the front ^ 1 , 5 ^ w i l l remain a shock (ie fiis = 2.0) although i t s v e l o c i t y w i l l decay since It is of the blast wave type. Once the unknown wave i s i d e n t i f i e d as a shock, a great s i m p l i f i c a t i o n r e s u l t s f o r the a p p l i c a t i o n of the SWING probe. The probing shocks are c o l l i d i n g with a non luminous, adiabatic shock which has an unknown v e l o c i t y . Solving equation (21) f o r V ^ , the SWING probe predicts the front v e l o c i t y ( f i g . 28) of the "unknown" 82- . DISTANCE (cm) -84-F i g . 27 F r o n t V e l o c i t i e s f o r the H igh P r e s s u r e C o l l i s i o n s - 1 - o l |_ l I I I I 0 - 6 0 - 8 1-0 1-2 1-4 1-6 1-8 2 0 V|5 (km/sec) - 8 6 - . shock and thus the flow v e l o c i t y immediately behind i t from the Rankine-Hugonlot equations. The equation p r e d i c t i n g the v e l o c i t y of (^1*5*) i s — t v*s "= 7-nc-x-^  * ( ^ - V ) ^ - ( 4 7 ) This a p p l i c a t i o n of the SWING probe demonstrates Its usefulness f o r the study of non luminous phenomena. Again assuming reasonable values f o r the adiabatic constants (g 1=g2=S3=g5= 1 .67? g^=l , 4 ), the expected v e l o c i t i e s V 2 3 » and can be calculated as a function of f o r a shock-shock c o l l i s i o n ( f i g . 2 7 ) . The excellent agreement i s a further i n d i c a t i o n that ^ 1 , 5 ^ Is a shock and is another consistency check f o r the technique. Theoretically„ the p a r t i c l e v e l o c i t y and density can be found throughout the continuous region behind the front ^ 1 , 5 ^ . However th i s requires a va r i a b l e supply of strong shocks which the av a i l a b l e detonation shock d r i v e r was not able to d e l i v e r . Some a d d i t i o n a l information about the continuous region i n the unknown -87-flow can however be obtained f o r c e r t a i n p r a c t i c a l approximations,, Uing equation (42), where i n t h i s case g± = 1.67. Sz =1.4, and M 1 2 = 2 , 5 . then we obtain simplys Equation (48) y i e l d s the p a r t i c l e v e l o c i t y v a r i a t i o n through the i n t e r i o r of the T-tube flow In the low v e l o c i t y regime. The v a r i a t i o n i n shock v e l o c i t y through the region between the front ,5^and the luminous front is evident i n figure 26b) as the curvature of the ^ 4 , 5 ^ front, The measurements of the shock v e l o c i t y ( f i g . 29) are taken from the smear pictures using point to point numerical d i f f e r e n t i a t i o n of the distance and time measurements. The r e p r o d u c i b i l i t y of these p a r t i c l e v e l o c i t y measurements i s of the order of 0 . 0 5 km/sec which i s impressive when compared to other methods of 34-37 measuring flow v e l o c i t i e s The s o l i d l i n e s i n figure 29 are t h e o r e t i c a l ; predictions of the p a r t i c l e v e l o c i t y based upon the metod of c h a r a c t e r i s t i c s ^ and provide a further check on the SWING probe r e s u l t s . The s o l u t i o n proceeded i n time steps with the p a r t i c l e v e l o c i t y at the ^ 1 , 5 ^ shock front and the luminosity front being treated as known boundary conditions. The p a r t i c l e v e l o c i t y at the luminosity f r o n t was assumed proportional to the luminosity front v e l o c i t y i n the r a t i o which holds at -88-F i g . 29 P a r t i c l e V e l o c i t y i n the Cont inuous Region Behind the T-Tube F r o n t 2 0 D I S T A N C E (cm) - 8 9 -the 5 cm p o s i t i o n i n figur e 28. The s o l i d l i n e s are the values calculated along the shock t r a j e c t o r i e s ( f i g . 25) which i s where the p a r t i c l e v e l o c i t i e s are a c t u a l l y measured. The good agreement between theory and experiment i s taken as a t h i r d v a l i d i t y check f o r the SWING probe. In f a c t the t h e o r e t i c a l treatment i s only approximate since the p a r t i c l e v e l o c i t y at the luminosity front i s only estimated and may e a s i l y be out by 0.1 km/sec. The experimental points do show a systematic trend toward lower values than the t h e o r e t i c a l l i n e s . This might be explained by a) the approximate nature of the t h e o r e t i c a l r e s u l t s , b) an error of10^ i n the estimated value f o r tf* , or c) d i f f u s i o n e f f e c t s at the luminosity front causing the t h e o r e t i c a l model to be incorrect near the luminous fr o n t . When the probing shock reaches the luminosity fr o n t , i t suffers a sudden change i n v e l o c i t y ( f i g . 26). The luminosity front i s n a t u r a l l y expected to be a contact surface separating the shock heated gas from the discharge heated gas and The s t r i k i n g feature i s that a c l a s s i c a l contact surface (equation (49))is not able to account f o r the measured increase i n shock v e l o c i t y . Equation (49) y i e l d s an unphyslcal r e s u l t - namely i t predicts a large negative density behind the luminosity f r o n t . ( A V ci 1.4 km/sec, -90-v± ~ 2 km/sec„ ( ^ ) £ - 0 . 3 1 , and |? - 2 ) . Evidently the luminous front i s supported by an energy source and i s probably maintained by the JxB f o r c e 0 Such sources would cause the p a r t i c l e v e l o c i t y to become negative across the surface which could account f o r the increase i n the probing shock v e l o c i t y . The luminous front i s s i m i l a r to a subsonic r a d i a t i o n front which i s a heating and expansion zone where the gas i s suddenly accelerated i n a backward d i r e c t i o n . A subsonic r a d i a t i o n f r o n t ^ w i l l also drive a shock wave ahead of i t , the high pressure between the shock and the heat front being maintained by the thrust of the rearward ejected hot gas. This s i m i l a r i t y i s not s u r p r i s i n g since no assumption about the absorbing mechanism was made i n deriving the 44 r e s u l t s about the r a d i a t i o n fronts , other than assuming a heat source l o c a l i z e d i n a d i s c o n t i n u i t y f r o n t . This heat source could be provided by r a d i a t i v e absorption as well as by ohmic heating or heat conduction. -91-SUMMARY The e f f o r t s f o r t h i s thesis have been directed by three separate although related ideas. The f i r s t concept was the study of shock propagation into inhomo-geneous media. The problem has been formulated In a general manner to obtain f i r s t order t h e o r e t i c a l r e s u l t s (equations ( 2 1 ) and ( 3 0 ) ) . F l u i d effects due to external sources have been treated extensively without actual examination of the mechanics of these sources. The theory of shock propagation into Inhomogeneous media has been extended to include v a r i a t i o n s i n the i n i t i a l pressure, p a r t i c l e v e l o c i t y , and energy sources appearing at the fron t . Future experimental work should t r y to v e r i f y the t h e o r e t i c a l equations and as well to s e t t l e the discrepancy between t h i s work and Rehm's work regarding the i n t e r a c t i o n of laser r a d i a t i o n with shock fronts. The second concept was the development of the SWING probe from the equations governing shock propagation into inhomogeneous regions. The SWING probe i n f e r s var-iatio n s i n the f l u i d parameters ahead of a probing shock from the va r i a t i o n s i n the shock v e l o c i t y . I t i s a p a r t i c u l a r l y e f f e c t i v e means f o r measuring flow v e l -o c i t i e s . Pressure, density, p a r t i c l e v e l o c i t y , and l o c a l energy input at the edge of an unknown flow can be determined from the measurement of the relevant fronts upon c o l l i s i o n of the probing shock with the edge of - 9 2 -the unknown flow, The steady v a r i a t i o n of the v e l o c i t y of strong probing shocks reveals d e t a i l s of the l o c a l v e l o c i t y and density d i s t r i b u t i o n s inside the unknown flow f i e l d . The t h i r d concept re s u l t s from the a p p l i c a t i o n of the SWING probe to the flow f i e l d produced by a T-tube. The SWING probe measurements have confirmed i n part the Q accepted understanding of the high and low v e l o c i t y regimes. In addition, very accurate flow v e l o c i t y measure-ments of the entire flow f i e l d have been obtained. When in t e r p r e t a t i n g the measurements i n terms of flow f i e l d s supported by energy sources, the high v e l o c i t y regime can be compared to a breakdown wave and the low v e l o c i t y regime i s l i k e a shock driven by a subsonic r a d i a t i o n f r o n t . The flow v e l o c i t y of a T-tube plasma has been measured with t y p i c a l accuracies of .1 km/sec which i s about 10% of the v e l o c i t y of the probing shock. In conclusion, t h i s probing technique c a l l s f o r more accurate methods of measuring front v e l o c i t i e s , and f o r the developement of shock drivers capable of de l i v e r i n g probe shocks v/lth a wide range of Mach numbers. - 9 3 -HEFERENCES 1. C.K.Chu and R.A,Gross Advances i n Plasma Physics 2 , 1 3 9 ( 1 9 6 9 ) 2 . R.A.Gross Rev. of Mod. Physics 2 7 , 7 2 4 ( 1 9 6 5 ) 3 . R.Courants and K.O.Friedricks "Supersonic Flow and Shock Waves" (Interscience Plublishers, New York 1948) 4 . L.Landau and E . 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Fluids 6 , 1 0 3 5 ( 1 9 6 3 ) 4 4 . B.Ahlborn andW.Zuzak Can. J. Phys. 47_, 1 8 0 9 ( 1 9 6 9 ) 4 5 . G.J.Pert J . Phys. D 3_»203(1970) 46. C.K.Chu Phys. Fluids £, 1 3 4 9 ( 1 9 6 4 ) 4 7 . J.P.Huni UBC Plasma Physics Lab. Report No. 6 (1969) 48. J.P.Huni Ph.D. Thesis UBC ( 1970) 49. J.D.Strachan, B.Ahlborn,and J.P.Huni Phys, Fluids ( i n press) - 5 0 . W.D.Hayes "The Basic Theory of Gasdynamic Discontin-u i t i e s " V ol. I l l , High Speed Aerody. and Jet Prop.- Fund, of Gas Dyn. , ed. H. Emmons? Princeton Univ. Press, 1 9 5 8 , pgs. 4 l 6 - 4 8 l . -96-APPENDIXi APPARATUS Following the straightfoward nature of the experiment, the apparatus and experimental setup are' very simple ( f i g . 2 0 ) , Most of the apparatus are i n common use i n our laboratory and were not b u i l t e s p e c i a l l y f o r t h i s experiment, a)THE SMEAR CAMERAt The smear camera^ allows f o r a continuous time record of one-dimensional luminous events. The camera ( f i g , 30) consists of a s l i t , a r o t a t i n g f r o n t - s i l v e r e d s p h e r i c a l mirror, and a recording surface (usually a f i l m ) . As the ni'irror rotates, the image sweeps across the f i l m with a speed %0O L , where OJ i s the angular speed of the mirror and L i s the mirror-t o - f i l m distance. An e l e c t r o n i c c i r c u i t , monitoring the rotor speed gives convenient t r i g g e r i n g f a c i l i t i e s . "The sweep speed c a l i b r a t i o n was obtained to absolute accuracy of 0.5% and a r e l a t i v e accuracy of 0 , 0 5 ^ . This was achieved by monitoring the period of the revolution of the rotor at the time the t r i g g e r pulse i s given out, A 5^5-A Tektronix oscilloscope i n conjunction with a type 7 8 l-A Dumont time mark generator was used to measure the period to four s i g n i f i c a n t f i g u r e s . The r e p r o d u c i b i l i t y of the camera electronics was found to be about 0„05%. The sweep speed was calculated from the period and the film-to-mirror distance ( L )• I t i s through t h i s distance that the 0.5% error i s introduced since i s known only Figure 30 General Layout of the Smear Camera -98-47 to within 0 , 5 cm f o r a t o t a l distance of 91 cm". The s p a t i a l r e s o l u t i o n of the camera is ontained from cm c a l i b r a t i o n marks on the shock tube, and therefore d i r e c t c a l i b r a t i o n on the f i l m was possible. The accuracy of the s p a t i a l r e s o l u t i o n was determined by the accuracy with which the markers were placed { Oi 0 . 2 mm). The v a r i a t i o n i n the magnification i n the image across the cen t r a l 2 cm of the s l i t i s of the order of 1% b) THE MICROSCOPE! The p o s i t i o n and time readings of the fron t were recorded from the smear pictures (polaroid 3000 ASA fil m ) using a dual feed t r a v e l l i n g microscope (Carl Z e i s s ) . The dual feed micrometers are capable of measuring to 10"" 3 Cm i n perpendicular d i r e c t i o n s on the f i l m . This implies that r e l a t i v e p o s i t i o n measurements are made to the order of 10"" ^  cm while r e l a t i v e time measurements are made to the order of 10 microseconds. The a c t u a l l i m i t a t i o n on the accuracy of the measurements (see the error bars i n fi g u r e 2 9 ) i s the width of the luminous front as i t appears on the f i l m (of the order of 3 x 10~3cm) and the percent error i n front v e l o c i t y depends of the distance i n t e r v a l over which the v e l o c i t y i s being measured. The X and T axes on the f i l m were a l l i g n e d with the perpendicular micrometers by running one micrometer feed along the cm marders on the f i l m . An absolute error of about 0 . 0 2 km/sec i s expected from misalignment due to d i f f e r i n g luminous i n t e n s i t y along -99-the cm markers at d i f f e r e n t times. c) THE DELAY UNITSs The extremely good r e p r o d u c i b i l i t y of the probing c o l l i s i o n s (figure 20) was obtained by' c a r e f u l l y arranging a series of delay u n i t s . F i r s t the pulse from the smear camera must be used to t r i g g e r a l l other phenomena since the mirror must be at just the r i g h t p o s i t i o n and v e l o c i t y to record the luminous phenomena focused on the entrance s l i t . The pulse from the smear camera occurs of the order of 1 millisecond before the mirror i s i n p o s i t i o n to record an event and i s v a r i a b l e . It i s used to t r i g g e r the detonation i n the shock tube d r i v e r . The detonation collapses onto the diaphragm which bursts sending a shock into the t e s t gas. The shock t r a v e l s about 1 ,5 meters downstream before i t arr i v e s i n the f i e l d of view of the smear camera. There i s a j i t t e r of the order of 5 - 1 0 microseconds of the p o s i t i o n of the shock on the f i l m (this i s unimporant since the t o t a l w r i t i n g time i s of the order of a hundred microseconds). This j i t t e r occurs as a r e s u l t of the j i t t e r i n the pulse fron the smear camera {1% of the t o t a l delay time which i s about 1 m i l l i s e c o n d ) . The r e l a t i v e j i t t e r i n the a r r i v a l of the shock once the detonation has been triggered i s only about 1 microsecond. The capacitor discharge ( t r i g g e r i n g the detonation) i s picked up by a Rogowski c o i l and i s used to t r i g g e r the T-tube discharge. In t h i s manner the T-tube discharge - 1 0 0 -i s triggered reproducibly r e l a t i v e to the shock generator* The Rogowski c o i l pulse triggers a Tektronix 162 waveform generator producing a long sawtooth wave which i s fed into a Tektronix 163 pulse generator. The pulse generator y i e l d s a pulse a f t e r a f r a c t i o n of the sawtooth wave duration. This pulse i s used to t r i g g e r the T-tube discharge a f t e r a set delay which ensures a c o l l i s i o n i n the f i e l d of view of the smear camera, d) THE SHOCK TUBE: The d r i v e r consists of a large detonation chamber f i t t e d with a front plate designed to adapt to 48 a shock tube . The detonation converges onto axis and bursts a Mylar diaphragm connecting the chamber to the shock tube. The shock tube i s constructed of 2.5 cm diameter pyrex tubing and i s about 1.5 meters long. This type of shock d r i v e r has excellent t r i g g e r i n g properties (due mainly to the very reproducible bursting of the diaphragm), however a xfide range of shock Mach numbers are not a v a i l a b l e f o r any s p e c i f i c tes gas pressure and there are very high percentage losses of shock heated test gas from the shock heated region due to boundary layer e f f e c t s (which are large due to the small tube diameter) and nonuniform pressure d i s t r i b u t i o n s i n the d r i v e r (across the tube diameter) -101-e) THE T-TU3E DISCHARGE CHAMBER- The T-tuhe electrodes are i n the standard configuration ( f i g . 3 D . The actual gap i s about 3 mm wide. The backstrap i s 0,7cm from the discharge and i s 2 . 5 cm wide and 0.2 cm thick copper. The discharge electrodes are 0 , 5 cm wide centered at the end of the shock tube. The chamber i s made out of l u c i t e and i s machined :to f i t r i g h t onto the end of the pyrex shock tube. The discharge chamber accumulates black deposites due to burnt b i t s of Mylar from the diaphragm at the shock d r i v e r end of the tube. f ) THE CAPACITOR BANK: The 60 microfarad capacitor bank ( f i g . 3 2 ) was connected as a lumped delay l i n e of impedance 0.7 ohms. This provided a constant current of approximately 5 kA with a pulse length of 80 microseconds f o r a charging voltage of 8 kV. The T-tube gap was i n i t i a t e d by a series spark gap triggered by a Thyratron/Theophanus u n i t . A probe r e s i s t o r plus avometer are used to give the true voltage since a meter before the charging r e s i s t o r must include the voltage drop across the charging r e s i s t o r . -102-F i g . 31 C o n s t r u c t i o n D e t a i l s o f the T-Tube Discharge Chamber channe l made to f i t pyrex shock tube e l e c t r o (Copper) 0 r i n g grove b a c k s t r a p SIDE VIEW n-103-F i g . 32 C a p a c i t o r Bank For T-Tube 40 w a t t , 2 £ 0 K ^ L c h a r g i n g R L L Sorenson 1020-30 power supp ly •C T C •C T C T-Tube e l e c t r o d and t r i g g e r spark gap L = 4 y\*h C = 10 R = 1 JX I n t e g r a t e d Rowgowski C o i l P i ckup (20^ sec/cm) ISSSSSBBSSII -104- , APPENDIX B - ANOTHER DERIVATION OF \.^U|) The phy s i c a l s i t u a t i o n corresponding to the p a r t i a l d e r i v a t i v e , [ , i s a shock crossing an i n - . f i n i t e s s i m a l d i s c o n t i n u i t y across which only the i n i t i a l p a r t i c l e v e l o c i t y changes. We note that U, i s the p a r t i c l e v e l o c i t y r e l a t i v e to the frame of the observer. Therefore, an analogous experimental s i t u a t i o n would be f o r an observer to watch the propagation of a shock t wave f i r s t from some i n e r t i a l frame at which U,s U, . The observer sees the shock propagate with some v e l o c i t y V while the shock strength i s 2 and the compression i s |r . Then, at time T= O , the observer changes to a d i f f e r e n t frame, one i n which the p a r t i c l e v e l o c i t y i s u.,st u.,' •+ clu., , where otu, i s n o n r e l a t i v i s t i c . To our observer, the flow of the shock wave appears as i n figure 15. Since nothing has happened to the flow i t s e l f while the observer has changed frames, the shock w i l l have the same strength parameters and /•?( , even though the shock v e l o c i t y i s d i f f e r e n t i n the new frame of reference. The change i n the shock v e l o c i t y i s duf , and therefore! The d e r i v a t i o n of equation ( 5 0 ) may at f i r s t seem rather q u a l i t a t i v e , however the d e r i v a t i o n i s coupled very strongly to the d e f i n i t i o n of v e l o c i t i e s and i t i s v a l i d to change the p a r t i c l e v e l o c i t y i n the above manner. - 1 0 5 -F i g . 15 I n t e r a c t i o n Scheme f o r the Second t/me D e r i v a t i o n o f ^ 3 u) 4-o " distance 3 ( A - 1 0 6 -However equation ( 5 0 ) can not be used i n equation ( 3 0 ) since equation ( 3 0 ) describes the change i n shock v e l o c i t y when the shock passes through an inhomogeneous region which features l o c a l v a r i a t i o n s In the f l u i d parameters. The p a r t i c l e v e l o c i t y as described by the i n t e r a c t i o n scheme i n fi g u r e 15 does not change l o c a l l y but changes the same over the en t i r e flow f i e l d . Therefore equation ( 5 0 ) i s a v a l i d c o e f f i c i e n t but i t does not apply i n the sit u a t i o n s which we are considering when we write equation ( 3 0 ) . 

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